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For example, in Figure 3, the set of treatments (Γ) and the set of rootstocks (Υ) are randomized to the set of trees (Ω).

Decomposition tables for experiments. II. Two--one randomizations

Unrandomized-inclusive randomizations in Example 1: rootstocks are randomized to trees in the initial experiment; in the superimposed experiment, treatments are randomized to trees taking account of the al location of rootstocks; B denotes Blocks.

Decomposition tables for experiments. II. Two--one randomizations

The initial experiment in Example 10 in is a randomized complete-block design to investigate cherry rootstocks: there are three blocks of ten trees each, and there are ten types of rootstock.

Decomposition tables for experiments. II. Two--one randomizations

This “square” is a 3 × 10 rectangle whose rows correspond to Blocks and columns to Rootstocks.

Decomposition tables for experiments. II. Two--one randomizations

Each of the ﬁve treatments occurs twice in each Block (row), while their disposition in Rootstocks (columns) is that of a balanced incomplete-block design.

Decomposition tables for experiments. II. Two--one randomizations

The sets of ob jects for this experiment are trees, rootstocks and treatments.

Decomposition tables for experiments. II. Two--one randomizations

The eﬃciency factors for the structure on treatments in relation to the joint decomposition of trees and rootstocks are derived from the extended Youden square.

Decomposition tables for experiments. II. Two--one randomizations

In this experiment VΓ ∩ V ⊥ 0 is not orthogonal to VΥ ∩ V ⊥ 0 , because the Viruses source is not orthogonal to Rootstocks.

Decomposition tables for experiments. II. Two--one randomizations

In particular, the Viruses source is not orthogonal to Trees[Blocks] ⊲ Rootstocks .

Decomposition tables for experiments. II. Two--one randomizations

The eﬃciency factors are recorded in the decomposition in Table 2, which shows that the Viruses source is partly confounded with both Rootstocks and the part of Trees[Blocks] that is orthogonal to Rootstocks. A consequence of this is that four Rootstocks degrees of freedom cannot be separated from Virus differences.

Decomposition tables for experiments. II. Two--one randomizations

However, there are ﬁve Rootstocks degrees of freedom that are orthogonal to Virus differences.

Decomposition tables for experiments. II. Two--one randomizations

Further, while the Viruses source has ﬁrst-order balance in relation to Rootstocks, the reverse is not true.

Decomposition tables for experiments. II. Two--one randomizations

The randomizations are independent, being carried out at different times and with the later one taking no account of the earlier one except to force fertilizers to be orthogonal to rootstocks.

Decomposition tables for experiments. II. Two--one randomizations

Consider the cherry rootstock experiment in Example 1.

Decomposition tables for experiments. II. Two--one randomizations

Similarly, the rootstocks tier decomposes VΥ into sources Mean and Rootstocks of dimensions 1 and 9.